Binary classifier that minimizes false positive error
There is no predefined classifier for any problem. Two main features of a classifier are

its cost function and
its corresponding weight update formula.

Since your problem statement requires a huge cost for falsely classifying a particular class one approach will be.

You have to define a cost function that will penalize hugely for misclassifying for that class only. So your cost function will be $J$ and $J'$ put together. You can look up the cost function of a logistic classifier to see how two separate cost functions are merged together here.

The second approach can be (assuming you are using supervised learning), the learning rate $\alpha$ for both the classes should be different. The larger learning rate will be for the one which is the more important class, since you don't want to classify it improperly (increasing $\alpha$ compared to the other class will reduce or risk of misclassifying it). The exact learning rate depends from case to case.


Thus, I have tailored the two main features of the classifier to solve this problem:

The cost function.

The weight update scheme (by changing learning rate for different cases).